[tex]xy=1\iff (r\cos\theta)(r\sin\theta)=r^2\sin\theta\cos\theta=\dfrac12r^2\sin2\theta=1[/tex]
[tex]\implies r^2=2\csc2\theta[/tex]
[tex]\implies r=\sqrt{2\csc2\theta}[/tex]
where we ignore the second root and use the convention that [tex]r>0[/tex].
